For what reasons might the immune system begin to attack a persons newly replaced organ?

It's complicated: I assume you mean a transplanted organ.The immune system is essentially trained to attack anything that is not "self." this includes viruses, etc., but also organs from other people. Donors and recipients of transplants are "matched" to try and minimize the differences, but unless it's identical twins, the immune system will eventually catch on. Immune suppressants help prevent this reaction.